David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jonathan Jenkins Ichikawa
Jack Alan Reynolds
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Synthese 170 (1):33 - 70 (2009)
David Hilbert’s early foundational views, especially those corresponding to the 1890s, are analysed here. I consider strong evidence for the fact that Hilbert was a logicist at that time, following upon Dedekind’s footsteps in his understanding of pure mathematics. This insight makes it possible to throw new light on the evolution of Hilbert’s foundational ideas, including his early contributions to the foundations of geometry and the real number system. The context of Dedekind-style logicism makes it possible to offer a new analysis of the emergence of Hilbert’s famous ideas on mathematical existence, now seen as a revision of basic principles of the “naive logic” of sets. At the same time, careful scrutiny of his published and unpublished work around the turn of the century uncovers deep differences between his ideas about consistency proofs before and after 1904. Along the way, we cover topics such as the role of sets and of the dichotomic conception of set theory in Hilbert’s early axiomatics, and offer detailed analyses of Hilbert’s paradox and of his completeness axiom (Vollständigkeitsaxiom).
|Keywords||David Hilbert Foundations of mathematics Mathematical existence Logicism Richard Dedekind Set theory Axiomatics Mathematical logic Real number system Paradoxes Georg Cantor Consistency proofs Models|
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References found in this work BETA
Bertrand Russell (1919). Introduction to Mathematical Philosophy. Dover Publications.
Bertrand Russell (1903). Principles of Mathematics. Cambridge University Press.
Paolo Mancosu (ed.) (1998). From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s. Oxford University Press.
Jean Van Heijenoort (ed.) (1967). From Frege to Gödel. Cambridge, Harvard University Press.
Colin McLarty (2006). Emmy Noether's “Set Theoretic” Topology: From Dedekind to the First Functors. In José Ferreirós Domínguez & Jeremy Gray (eds.), The Architecture of Modern Mathematics: Essays in History and Philosophy. Oxford University Press 187--208.
Citations of this work BETA
Erich H. Reck (2013). Frege, Dedekind, and the Origins of Logicism. History and Philosophy of Logic 34 (3):242-265.
Eduardo N. Giovannini (2016). Bridging the Gap Between Analytic and Synthetic Geometry: Hilbert’s Axiomatic Approach. Synthese 193 (1):31-70.
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